Group #34208159 — subgroups

Subgroups — #34208159

Order 20224.

292 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765195	C₂	1	1	yes	maximal, characteristic, complemented
2	#23763438	C₂	1	2	yes	maximal
3	#23761512	C₂	1	2	yes	maximal, complemented
4	#23761504	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#2112	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	C₂²	1	2	yes	—
7	C₂×#14579097	C₂²	1	2	yes	—
8	#19098527	—	2	2	no	—
9	#19098393	—	2	2	no	—
10	#19098328	—	2	2	no	—
11	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₁₆×C₇₉	—	2	2	no	—
15	C₂×#11929608	—	2	2	no	—
16	C₂×#11929608	—	2	2	no	—
17	#14579097	—	2	2	no	—
18	#14579097	—	2	2	no	—
19	#14579097	—	2	2	no	—
20	#14579097	—	2	2	no	—
21	C₂²⋊(C₇₉⋊C₈)	—	4	4	no	—
22	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	—	2	1	no	—
24	C₂²×C₈×C₇₉	—	2	2	no	—
25	C₂²×C₈×C₇₉	—	2	2	no	—
26	C₂²×C₈×C₇₉	—	2	1	no	—
27	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	M₃₂×C₇₉	—	2	2	no	—
30	M₃₂×C₇₉	—	2	2	no	—
31	C₂×C₁₆×C₇₉	—	2	1	no	—
32	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
33	#11929608	—	4	2	no	—
34	#11929608	—	4	2	no	—
35	#11929608	—	4	2	no	—
36	#11929608	—	4	2	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₂×C₇₉⋊C₈	—	4	2	no	—
39	C₂⁴×C₇₉	Q₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	—	2	1	no	—
41	C₂²×C₄×C₇₉	—	2	2	no	—
42	C₂²×C₄×C₇₉	—	2	1	no	—
43	C₂²×C₄×C₇₉	—	2	2	no	—
44	C₂²×C₄×C₇₉	Q₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	—	2	1	no	—
47	C₂×C₈×C₇₉	—	2	2	no	—
48	C₂×C₈×C₇₉	—	4	1	no	—
49	C₂×C₈×C₇₉	—	4	1	no	—
50	C₂×C₈×C₇₉	—	4	2	no	—
51	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₉	#41	1	2	yes	—
54	C₂×C₈×C₇₉	#41	1	2	yes	—
55	C₁₆×C₇₉	—	2	2	no	—
56	C₁₆×C₇₉	—	4	1	no	—
57	C₂³×C₇₉	—	2	1	no	—
58	C₂³×C₇₉	—	2	2	no	—
59	C₂³×C₇₉	—	4	2	no	—
60	C₂³×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₉	—	2	1	no	—
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	—	4	1	no	—
64	C₂×C₄×C₇₉	—	4	1	no	—
65	C₂×C₄×C₇₉	—	4	2	no	—
66	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#123	1	2	yes	—
68	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
69	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
70	C₈×C₇₉	—	4	1	no	—
71	C₈×C₇₉	#123	1	1	yes	characteristic
72	C₈×C₇₉	#124	1	1	yes	characteristic
73	C₈×C₇₉	#126	1	2	yes	—
74	C₇₉⋊C₈	—	8	2	no	—
75	C₇₉⋊C₈	—	8	2	no	—
76	C₂²×C₇₉	—	2	2	no	—
77	C₂²×C₇₉	—	2	4	no	—
78	C₂²×C₇₉	—	4	1	no	—
79	C₂²×C₇₉	—	4	1	no	—
80	C₂²×C₇₉	—	4	2	no	—
81	C₂²×C₇₉	—	4	1	no	—
82	C₂²×C₇₉	#467	1	1	yes	characteristic
83	C₂²×C₇₉	#484	1	2	yes	—
84	C₄×C₇₉	—	4	1	no	—
85	C₄×C₇₉	#397	1	1	yes	characteristic
86	C₄×C₇₉	#469	1	1	yes	characteristic
87	C₄×C₇₉	#482	1	2	yes	—
88	#13498	—	79	1	no	maximal, Sylow
89	C₂×C₇₉	—	2	1	no	—
90	C₂×C₇₉	—	2	2	no	—
91	C₂×C₇₉	—	4	2	no	—
92	C₂×C₇₉	#2076	1	1	yes	characteristic
93	#2161	—	79	1	no	—
94	#2112	D₇₉	1	1	yes	characteristic
95	#1810	—	79	2	no	—
96	#1388	—	79	2	no	—
97	#1385	—	79	1	no	—
98	C₇₉	#13498	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
100	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
101	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
102	C₂×#108	—	79	2	no	—
103	C₂×#108	—	79	2	no	—
104	#407	—	158	2	no	—
105	#368	—	158	2	no	—
106	C₂²⋊C₁₆	—	2	2	no	—
107	#341	—	158	2	no	—
108	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
109	C₂×M₁₆	—	158	2	no	—
110	C₂×M₁₆	—	158	2	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	—	2	2	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
121	#108	—	158	2	no	—
122	#108	—	158	2	no	—
123	#108	—	158	2	no	—
124	#108	—	158	2	no	—
125	C₂²⋊C₈	—	316	4	no	—
126	C₂⁴	Dic₃₁₆	1	1	yes	characteristic
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	Dic₃₁₆	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
133	M₁₆	—	316	2	no	—
134	M₁₆	—	316	2	no	—
135	M₁₆	—	316	2	no	—
136	M₁₆	—	316	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	316	2	no	—
143	C₂×C₈	—	316	2	no	—
144	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
145	C₂×C₈	#11929623	1	2	yes	—
146	C₂×C₈	#11929623	1	2	yes	—
147	C₂×C₈	#11929624	1	1	yes	characteristic
148	C₁₆	—	2	2	no	—
149	C₁₆	—	4	1	no	—
150	C₂³	—	2	2	no	—
151	C₂³	—	2	1	no	—
152	C₂³	—	4	2	no	—
153	C₂³	C₂×Dic₃₁₆	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	#14579135	1	1	yes	characteristic
160	C₂×C₄	#14579145	1	2	yes	—
161	C₂×C₄	#14579183	1	1	yes	characteristic
162	C₂×C₄	#14579189	1	2	yes	—
163	C₈	—	4	1	no	—
164	C₈	—	632	2	no	—
165	C₈	—	632	2	no	—
166	C₈	#14579142	1	1	yes	characteristic
167	C₈	#14579148	1	1	yes	characteristic
168	C₈	#14579154	1	2	yes	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	1	no	—
175	C₂²	#19098804	1	1	yes	characteristic
176	C₂²	#19098899	1	2	yes	—
177	C₄	—	4	1	no	—
178	C₄	#19098488	1	1	yes	characteristic
179	C₄	#19098816	1	1	yes	characteristic
180	C₄	#19098888	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	2	1	no	—
183	C₂	—	4	2	no	—
184	C₂	#23764828	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765195

C₂

yes

maximal, characteristic, complemented

#23763438

C₂

yes

maximal

#23761512

C₂

yes

maximal, complemented

#23761504

C₂

yes

maximal, characteristic, complemented

C₇₉×#2112

C₂

yes

maximal, characteristic

C₂×#14579097

C₂²

yes

—

C₂×#14579097

C₂²

yes

—

#19098527

—

#19098393

—

#19098328

—

C₂³×C₈×C₇₉

C₂²

yes

characteristic

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₉

—

—

—

—

—

—

—

C₂²⋊(C₇₉⋊C₈)

—

C₂³×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

M₃₂×C₇₉

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

—

—

—

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

Q₁₆

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

Q₁₆

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₈×C₇₉

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#467

yes

characteristic

C₂²×C₇₉

#484

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#13498

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

D₇₉

yes

—

—

—

C₇₉

#13498

yes

characteristic, complemented, Sylow

C₂³×C₈

C₂×D₇₉

yes

characteristic

100

C₂×M₃₂

C₂×D₇₉

yes

characteristic

101

C₂×M₃₂

C₂×D₇₉

yes

characteristic

102

C₂×#108

—

103

C₂×#108

—

104

#407

—

158

—

105

#368

—

158

—

106

C₂²⋊C₁₆

—

107

#341

—

158

—

108

C₂³×C₄

D₃₁₆

yes

characteristic

109

C₂×M₁₆

—

158

—

110

C₂×M₁₆

—

158

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

D₃₁₆

yes

characteristic

116

C₂²×C₈

C₂²×D₇₉

yes

characteristic

117

M₃₂

—

118

M₃₂

—

119

C₂×C₁₆

—

120

C₂×C₁₆

C₂²⋊D₇₉

yes

—

121

#108

—

158

—

122

#108

—

158

—

123

#108

—

158

—

124

#108

—

158

—

125

C₂²⋊C₈

—

316

—

126

C₂⁴

Dic₃₁₆

yes

characteristic

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

Dic₃₁₆

yes

characteristic

132

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

133

M₁₆

—

316

—

134

M₁₆

—

316

—

135

M₁₆

—

316

—

136

M₁₆

—

316

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

316

—

143

C₂×C₈

—

316

—

144

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

145

C₂×C₈

#11929623

yes

—

146

C₂×C₈

#11929623

yes

—

147

C₂×C₈

#11929624

yes

characteristic

148

C₁₆

—

149

C₁₆

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

C₂×Dic₃₁₆

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

#14579135

yes

characteristic

160

C₂×C₄

#14579145

yes

—

161

C₂×C₄

#14579183

yes

characteristic

162

C₂×C₄

#14579189

yes

—

163

C₈

—

164

C₈

—

632

—

165

C₈

—

632

—

166

C₈

#14579142

yes

characteristic

167

C₈

#14579148

yes

characteristic

168

C₈

#14579154

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#19098804

yes

characteristic

176

C₂²

#19098899

yes

—

177

C₄

—

178

C₄

#19098488

yes

characteristic

179

C₄

#19098816

yes

characteristic

180

C₄

#19098888

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

#23764828

yes

characteristic

	subgroup	G/H
center	C₄	#19098816
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2112	C₂
socle	C₂×C₇₉	#2076
prime radical	C₂⁴×C₇₉	Q₁₆
hypercenter	#2112	D₇₉
hypocenter	C₇₉	#13498
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₄

#19098816

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2112

C₂

socle

C₂×C₇₉

#2076

prime radical

C₂⁴×C₇₉

Q₁₆

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
1. #23765195	C₂	semidirect
3. #23761512	C₂	semidirect
4. #23761504	C₂	semidirect
98. C₇₉	#13498	semidirect

H ≅ G/N

type

1. #23765195

C₂

semidirect

3. #23761512

C₂

semidirect

4. #23761504

C₂

semidirect

98. C₇₉

#13498

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23765195	2
#23763438	2
#23761512	2
#23761504	2
C₇₉×#2112	2
#13498	79

subgroup

index

C₇₉×#2112

Subgroups — #34208159

Conjugacy classes of subgroups

Complements

Maximal subgroups